Finding number of roots of a complex equation using rouche's theorem

In summary, Rouche's theorem is a mathematical theorem used to determine the number of roots of a complex polynomial in a given region of the complex plane. It involves comparing the number of roots of a complex polynomial to that of a simpler polynomial within the same region. The polynomial must be complex and the region must be simply connected for the theorem to be applicable. It cannot be used for all complex polynomials and has limitations in determining the exact location of the roots and may not work for more complex regions.
  • #1
abhijeet.26
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Homework Statement


determine the number of roots, counting multiplicities, of the equation z^7-5*z^3+12=0
in side the annulus 1<=|z|<2


Homework Equations

use rouche's theorem



The Attempt at a Solution

 
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  • #2

1. What is Rouche's theorem?

Rouche's theorem is a mathematical theorem that can be used to determine the number of roots of a complex polynomial in a given region of the complex plane.

2. How does Rouche's theorem work?

Rouche's theorem involves comparing the number of roots of a complex polynomial within a given region to the number of roots of a simpler polynomial within the same region. By comparing the two polynomials, we can determine the number of roots of the more complex polynomial.

3. What are the requirements for using Rouche's theorem?

In order to use Rouche's theorem, the polynomial must be complex and the region in the complex plane must be simply connected (meaning there are no holes or cuts in the region).

4. Can Rouche's theorem be used for any complex polynomial?

No, Rouche's theorem can only be used for polynomials that are analytic (meaning they have a derivative) and have no poles (values that make the polynomial undefined).

5. Are there any limitations to Rouche's theorem?

Rouche's theorem has limitations in that it cannot determine the exact location of the roots, only the number of roots within a given region. Additionally, it may not work for more complex or unusual regions in the complex plane.

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